How might spatial nonuniformity of dose in a homogeneous biological system affect its total response?

The total response of a homogeneous biological system to a fixed total dose of a biological agent is modeled by dividing the system into N cubical voxels, each of which can be associated with an individual dose D(n) and an individual response R(n) =F(D(n)). Among the results shown are the following: A. (Voxel Theorem). Let the average dose D(avg) be held fixed as the dose distribution is shifted from uniform u to arbitrary a. Then, if F' > or = 0 over [D(min), D(max)] and R = summation operator (n = 1)(N) R(n), a sufficient condition that NF(D(avg)) = R(u) < or = R(a) is that F be a concave-upwards function of dose; that is, F" > or = 0 over [D(min), D(max)]. B. If F' is constant over [D(min), D(max)], then R(a) = R(u). That is, the total response is a function of D(avg) only. The applications of these (and other) results are illustrated by examples from bioelectromagnetics.